Imagine two baskets, each one with two stones in it. Mathematics tells you that if you drop the contents of one basket into the other, you will end up with a basket with four stones inside. You do the experiment and find that it works. What's mysterious about it?

Math "works" because, contrary to what many people tend to think, it doesn't do anything. Math never tells you what you don't already know, so it's not surprising that it never lies.

Math never tells you what you don't already know, so it's not surprising that it never lies.

Math never gives you information beyond that contained within the given axioms and the given system of inference. At the same time, it can certainly show you things you don't know (things that aren't yet apparent to you) about the information contained within your axioms + inference system.

For example, given standard logic and the axioms of Euclidean geometry, there is no information in the derived Pythagorean theorem that is not already implicitly contained within the axioms given our rules of logical inference. At the same time, we would be remiss to say that a beginning student who is taught the rules of inference and the axioms of Euclidean geometry already knows the Pythagorean theorem. Deriving the theorem is a sort of unfolding of the collective system to make an implicit consequence of it into an explicit statement, which amounts to a gain in knowledge.

Math never gives you information beyond that contained within the given axioms and the given system of inference. At the same time, it can certainly show you things you don't know (things that aren't yet apparent to you) about the information contained within your axioms + inference system.

I'm not sure what we mean when we say we don't know something; the concept of "knowing" is a bit elusive. For instance, you know that 2 is the square root of 4, but I doubt you knew that 111 is the square root of 12,321. Yet from a logical perspective there is just as much information in the statement SQRT(4) = 2 as in SQRT(12321) = 111; both are trivial facts, but the former seems more trivial to you than the latter. That difference in perception is what prompts people to think math has some magical power to tell us things about reality that are not trivial.

given standard logic and the axioms of Euclidean geometry, there is no information in the derived Pythagorean theorem that is not already implicitly contained within the axioms given our rules of logical inference. At the same time, we would be remiss to say that a beginning student who is taught the rules of inference and the axioms of Euclidean geometry already knows the Pythagorean theorem. Deriving the theorem is a sort of unfolding of the collective system to make an implicit consequence of it into an explicit statement, which amounts to a gain in knowledge.

Perhaps we can call it knowledge, but the fact remains that mathematics mystifies people only because we are not particularly smart. A fully conscious being who could see all the consequences of his knowledge would consider mathematics completely useless.

I don’t actually understand my question completely although some of the answers here have given me a better understanding. I seem to be able to get to an answer independent of the method I use to get there. This made me curious why it works so elegantly, I can also not “cheat” because values cancel out.

Another example of the sort he used can be taken from group theory. There are four simple axioms that define group. Another few definitions (including the definition of normal subgroup) will tell you what it means for a group to be finite and to be simple. In principle once you know that, you can come to know that the finite simple groups fall into four infinite families plus 26 sporadic groups. But the historical fact is that it took many thousands (millions?) of mathematician-hours of thinking to wring that knowledge out of the axioms and definitions.

(A thank-you to Matt Grime for the information in another thread that this classification is still in some doubt among certain group theorists. Maybe the final word has not yet been written on this topic.)

Matt can correct me if I'm wrong, but I have heard that the problem with this proof is just that it takes the equivalent of several books to lay it all out, and no one mathematician however powerful is able to get his mind completely around the whole thing. So in the traditional sense of statisfactory proof, it isn't, quite.

What they had was a group of theorists with overlapping parts of the proof, so each one could verify the one before him and after him.

Matt can correct me if I'm wrong, but I have heard that the problem with this proof is just that it takes the equivalent of several books to lay it all out, and no one mathematician however powerful is able to get his mind completely around the whole thing.- selfAdjoint

So I've heard. The Four-Color theorem is another one that a human mathematician can't get his mind completely around, it seems. Thus the necessity of a computer program. This makes me wonder if there is some practical way of programming a computer to check the classification theorem for finite simple groups. I don't know if anybody has tried to write a program to do that--probably not.

There exist basic proof checkers, but they are too cumbersome to be used. I suspect that it would also take far too long to translate statements into computer readable form for them to check big results for a while yet.

The main issue with CFSG is its size, several thousands of pages of papers. Basically, different sets of peopll took different intervals (10,000 to 20,000) and tried to find the simple groups with orders in that block. If anyone particulary cares, there are several infinite families of simple groups (alternating etc) and then there are the sporadic ones. If you are wondering how they managed to decide there were only finitely many exceptions then it boils down to the fact that 2*3=2+3+1. Or more accurately, that S_6 possesses an outer automorphism (and is unique amongst permutation groups becuase of this weird numerical fact) and this can be used to create all these simple groups.

If you wish to read more about this then I suggest searching for things by Conway, and Simon Norton.

In effect there is not *a* proof of the theorem, but there are lots of special cases considered, too many for some people's taste, but it has withstood scrutiny for many years now (and bear in mind one only needs 18 months to elapse after publication in an international journal to get a Clay prize).

As for the idea that the answer Repugno arrives at is independent of the method, well, that is part of how mathematics is defined. The methods prodice the same answer because if they didn't, they, by definition, wouldn't be equivalent and would therefore have at least one mistake in them, or something.

Although I must qualify that and say that is a modern view on mathematics. For many years (the immediate centuries prior to Gauss, and possibly even later) maths was treated as part of the natural philosophy school of thought, and the level of proof we require today wasn't recognized. In particular the modern axiomatic ideals, that something is what it does, didn't hold sway, so it was felt a mistake that certain equations didn't have roots. Initially, say, Cardano's formula was felt to be wrong because when applied to cubics with known roots it produced answers that required square roots of negative numbers to be used.

Now, we'd just look at the result and decide to open a new avenue of research, but then it was treated with suspicion.

I'll keep an eye out for Conway (for whom one of the sporadics is named, as I recall) and Norton. Years ago I came across a library book by Daniel Gorenstein, and I tried to work my way though it, but I didn't get very far into it before bogging down.